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The rare decays of a kaon into a pion and a charged lepton/antilepton pairproceed via a flavour changing neutral current and therefore may only beinduced beyond tree level in the Standard Model. This natural suppression makesthese decays sensitive to the effects of potential New Physics. The CPconserving $K\to\pi \ell^+\ell^-$ decay channels however are dominated by asingle photon exchange; this involves a sizeable long-distance hadroniccontribution which represents the current major source of theoreticaluncertainty. Here we outline our methodology for the computation of thelong-distance contributions to these rare decay amplitudes using lattice QCDand present the numerical results of the first exploratory studies of thesedecays in which all but the disconnected diagrams are evaluated. The domainwall fermion ensembles of the RBC and UKQCD collaborations are used, with apion mass of $M_{\pi}\sim 430\,\mathrm{MeV}$ and a kaon mass of $M_{K}\sim625\,\mathrm{MeV}$. In particular we determine the form factor, $V(z)$, of the$K^+\to\pi^+\ell^+\ell^-$ decay from the lattice at small values of$z=q^2/M_{K}^{2}$, obtaining $V(z)=1.37(36),\, 0.68(39),\, 0.96(64)$ for thethree values of $z=-0.5594(12),\, -1.0530(34),\, -1.4653(82)$ respectively.

First exploratory calculation of the long-distance contributions to the rare kaon decays $K\to\pi\ell^+\ell^-$